Spot the error: Which of the following Boolean algebra rules is incorrect under standard Boolean conventions?

Introduction / Context:
Boolean algebra follows a small set of identities that greatly simplify logic design. Recognizing invalid identities prevents design errors when reducing logic expressions.

Given Data / Assumptions:

• Standard Boolean symbols: + for OR, · for AND, and overbar for NOT.
• We assume variables are Boolean (0 or 1).
• We must identify the false identity.

Concept / Approach:
Key identities include identity, null, idempotent, and dominance laws. The correct forms are: 0 + A = A (identity for OR), 1 + A = 1 (dominance for OR), A + A = A (idempotent), and 1 · A = A (identity for AND). Any deviation from these is incorrect.

Step-by-Step Solution:
Check 0 + A = A → true (OR identity).Check 1 + A = 1 → true (OR dominance).Check A + A = A → true (idempotent law).Check 1 · x = 1 → false; correct is 1 · x = x (AND identity).

Verification / Alternative check:
Truth table for 1 · x: when x = 0, 1 · 0 = 0, not 1; when x = 1, 1 · 1 = 1. Therefore the expression equals x, not constant 1.

Why Other Options Are Wrong:
0 + A = A: a valid identity.1 + A = 1: a valid dominance law.A + A = A: a valid idempotent law.None of the above: incorrect because one rule is indeed wrong.

Common Pitfalls:
Confusing dominance vs. identity laws; mixing constant 1 behavior across AND and OR.

Final Answer:
1 . x = 1